Black Holes: Videos & Practice Problems

Black holes are incredibly dense objects with such strong gravitational pull that not even light can escape. The Schwarzschild radius defines the size of a black hole, calculated using the equation $2^G\frac{M}{c^2}$. Anything crossing the event horizon is trapped forever. For example, a black hole with a Schwarzschild radius of 120 AU has a mass approximately 6 billion times that of the Sun, illustrating the immense density and gravitational influence of black holes in the universe.

Black Holes

Black holes are intriguing astronomical objects characterized by their immense mass concentrated in a relatively small volume. For instance, a black hole can have a mass equivalent to billions of suns, yet occupy a space of only about 500 kilometers in diameter. This extreme density creates a gravitational pull so strong that not even light can escape, which is why they appear black. The concept of escape velocity is crucial here; to escape a black hole's gravitational pull, an object would need to exceed the speed of light, approximately \(3 \times 10^8\) meters per second, which is impossible according to the laws of physics.

One of the key equations related to black holes is the Schwarzschild radius, which defines the boundary around a black hole known as the event horizon. The event horizon marks the point beyond which nothing can escape the black hole's gravitational influence. The equation for the Schwarzschild radius (\(R_S\)) is given by:

\[ R_S = \frac{2GM_{BH}}{c^2} \]

In this equation, \(G\) represents the gravitational constant (\(6.67 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2\)), \(M_{BH}\) is the mass of the black hole, and \(c\) is the speed of light. To find the mass of a black hole in terms of solar masses, one can rearrange the equation to solve for \(M_{BH}\):

\[ M_{BH} = \frac{R_S c^2}{2G} \]

For practical calculations, it is essential to convert all measurements to SI units. For example, if the Schwarzschild radius is given as 120 astronomical units (AU), it must be converted to meters (1 AU = \(1.5 \times 10^{11}\) meters), resulting in a Schwarzschild radius of approximately \(1.8 \times 10^{13}\) meters.

Using this radius in the rearranged Schwarzschild equation allows astronomers to calculate the mass of the black hole. After performing the calculations, one might find that a black hole has a mass of about \(1.21 \times 10^{40}\) kilograms. To express this mass in terms of solar masses, where the mass of the sun is approximately \(2 \times 10^{30}\) kilograms, the final result could indicate that the black hole is around \(6 \times 10^9\) times the mass of the sun, illustrating the extraordinary density and gravitational influence of black holes within the cosmos.

Astronomers have found a small, but incredibly massive object at the center of our Milky Way galaxy, and suspect it is a black hole. A cloud of gas orbits this object at $1.8\times 10^5$ m/s every 78,500 years.
a) What is the mass of this alleged black hole?
b) How large is this black hole?

• A. (a) $3.45\times 10^{37}$ kg (b) $5.11\times 10^{10}$ m
• B. (a) $3.45\times 10^{37}$ kg (b) $7.1\times 10^{16}$ m
• C. (a) $1.92\times 10^{34}$ kg (b) $7.1\times 10^{16}$ m
• D. (a) $1.92\times 10^{34}$ kg (b) $5.11\times 10^{10}$ m

Answer: (A)

Replacing the sun with a black hole

In this scenario, we explore the effects of a black hole, equivalent in size to Earth, replacing the Sun. The gravitational forces at play involve two accelerations: the gravitational acceleration due to Earth, denoted as \( g_{\text{Earth}} \), which is approximately \( 9.8 \, \text{m/s}^2 \), and the gravitational acceleration due to the black hole, represented as \( g_{\text{BH}} \).

To determine the net acceleration experienced by objects on Earth's surface, we need to calculate the gravitational acceleration from the black hole using the formula:

\[ g_{\text{BH}} = \frac{G \cdot m_{\text{BH}}}{r^2} \]

Here, \( G \) is the gravitational constant, \( m_{\text{BH}} \) is the mass of the black hole, and \( r \) is the distance from the black hole to the Earth, which is the average distance from the Earth to the Sun, approximately \( 1.5 \times 10^{11} \, \text{m} \).

To find the mass of the black hole, we utilize the Schwarzschild radius equation, which relates the size of the black hole to its mass:

\[ r_s = \frac{2G \cdot m_{\text{BH}}}{c^2} \]

Rearranging this equation to solve for \( m_{\text{BH}} \) gives us:

\[ m_{\text{BH}} = \frac{r_s \cdot c^2}{2G} \]

Given that the Schwarzschild radius \( r_s \) is equal to the radius of Earth, approximately \( 6.37 \times 10^6 \, \text{m} \), and using the speed of light \( c \) as \( 3 \times 10^8 \, \text{m/s} \), we can substitute these values into the equation:

\[ m_{\text{BH}} = \frac{(6.37 \times 10^6) \cdot (3 \times 10^8)^2}{2 \cdot (6.67 \times 10^{-11})} \approx 4.30 \times 10^{33} \, \text{kg} \]

Now, substituting \( m_{\text{BH}} \) back into the equation for \( g_{\text{BH}} \), we find:

\[ g_{\text{BH}} = \frac{(6.67 \times 10^{-11}) \cdot (4.30 \times 10^{33})}{(1.5 \times 10^{11})^2} \approx 12.6 \, \text{m/s}^2 \]

To find the net gravitational acceleration \( g_{\text{net}} \), we subtract the gravitational acceleration due to Earth from that of the black hole:

\[ g_{\text{net}} = g_{\text{BH}} - g_{\text{Earth}} = 12.6 - 9.8 = 2.8 \, \text{m/s}^2 \]

This positive value indicates that objects on Earth's surface would experience an upward acceleration, suggesting they would be lifted off the surface due to the influence of the black hole. Thus, the net acceleration is \( 2.8 \, \text{m/s}^2 \), confirming that the gravitational pull of the black hole is sufficient to overcome Earth's gravity in this hypothetical scenario.

Here’s what students ask on this topic:

What is a black hole and how is it formed?

A black hole is an astronomical object with an extremely strong gravitational pull, so intense that not even light can escape from it. Black holes are formed when massive stars exhaust their nuclear fuel and undergo gravitational collapse. This collapse compresses the star's core to a point of infinite density known as a singularity, surrounded by an event horizon, the boundary beyond which nothing can escape. The size of a black hole is defined by its Schwarzschild radius, which depends on its mass.

What is the Schwarzschild radius and how is it calculated?

The Schwarzschild radius is the radius of the event horizon of a black hole, beyond which nothing can escape its gravitational pull. It is calculated using the equation:

$R_S=\frac{2GM}{c^2}$

where $G$ is the gravitational constant, $M$ is the mass of the black hole, and $c$ is the speed of light.

What happens to objects that fall into a black hole?

Objects that fall into a black hole cross the event horizon and are trapped forever. The gravitational pull is so strong that they cannot escape, even if they travel at the speed of light. As they approach the event horizon, they experience extreme tidal forces that can stretch and compress them, a process known as spaghettification. Once inside the event horizon, they move inexorably towards the singularity, where they are crushed to infinite density.

How do astronomers detect black holes if they emit no light?

Astronomers detect black holes by observing their effects on nearby objects and light. For example, they can detect the gravitational influence of a black hole on the orbits of nearby stars or gas clouds. Additionally, when matter falls into a black hole, it heats up and emits X-rays, which can be detected by telescopes. The presence of an event horizon can also be inferred from the absence of light from the region around the black hole.

What is the significance of the event horizon in a black hole?

The event horizon is the boundary around a black hole beyond which nothing can escape. It is significant because it marks the point of no return; once an object crosses this boundary, it is inevitably pulled into the black hole. The event horizon is also where the escape velocity equals the speed of light, making it impossible for anything, including light, to escape. This is why black holes appear black, as no light can reach an observer from within the event horizon.